Page 281 - Adaptive Identification and Control of Uncertain Systems with Nonsmooth Dynamics

Identification and Control of Hammerstein Systems With Hysteresis Non-linearity 283

and the inverse Preisach model is
−1
U = (μ ˆω) X = δX (18.20)

where
⎡ ⎤
1 0 0 ... 0
⎢ 1 1 0 ... 0 ⎥
⎢
⎥
ˆ ω = ⎢1 1 1 ... 0⎥ (18.21)
⎥
⎢
⎢ ⎥
⎣... ... ... ... ...⎦
1 1 1 ... 1
−1
and δ = (μ ˆω) .
Proof. Discretize the Preisach plane as Fig. 18.2D and consider the defi-
nition of Preisach operator γ α,β [u,ζ(α,β)] and (18.17), we construct that
all the weight of the cells in triangle Preisach plane are 1. Moreover, each
input u i is averagely assigned to the cells of the i-th horizon, so that we can
define one of the functions of the Preisach operator as γ α,β [u,ζ(α,β)] as

¯ γ =ˆω ˆ U (18.22)

where ¯γ indicates the Preisach operator γ α,β [u,ζ(α,β)] of one piecewise
monotonic section (e.g., Fig. 18.2C (a)), ˆω and ˆ U are defined as in (18.18)
and (18.21).
Rewrite Eq. (18.17) into matrices based on (18.18), (18.21), and
(18.22), then, ˆ X is obtained as

ˆ X = μ ¯γ = μ ˆω ˆ U (18.23)
Since ˆ U and ˆω are triangle matrices, their inverse matrices can be easily
obtained. Thus, the following fact holds

−1
μ = ˆ X ˆ U −1 ˆ ω . (18.24)
Moreover, Eq. (18.17) can be rewritten as

X = μ ˆωU (18.25)

so that the inverse Preisach model is obtained as follows:
−1
U = (μ ˆω) X = δX (18.26)

This completes the proof.